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Stochastic Forward–Backward Splitting for Monotone Inclusions

Author

Listed:
  • Lorenzo Rosasco

    (Università di Genova
    Istituto Italiano di Tecnologia and Massachusetts Institute of Technology)

  • Silvia Villa

    (Istituto Italiano di Tecnologia and Massachusetts Institute of Technology)

  • Bang Công Vũ

    (Istituto Italiano di Tecnologia and Massachusetts Institute of Technology)

Abstract

We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward–backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as well as almost sure convergence under weaker assumptions. For minimization problems, we recover rates matching those obtained by stochastic extensions of the so-called accelerated methods. Stochastic quasi-Fejér’s sequences are a key technical tool to prove almost sure convergence.

Suggested Citation

  • Lorenzo Rosasco & Silvia Villa & Bang Công Vũ, 2016. "Stochastic Forward–Backward Splitting for Monotone Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 388-406, May.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:2:d:10.1007_s10957-016-0893-2
    DOI: 10.1007/s10957-016-0893-2
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    References listed on IDEAS

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    1. DEVOLDER, Olivier, 2011. "Stochastic first order methods in smooth convex optimization," LIDAM Discussion Papers CORE 2011070, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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